85/VB02. problem / Math - Solved Math Problems

MR-85 / 221a. problem

Simplify the following expression:

$\frac{\sqrt{x} + \sqrt{y} - 1}{x + \sqrt{x y}} + \frac{\sqrt{x} - \sqrt{y}}{2 \sqrt{x y}} (\frac{\sqrt{y}}{x - \sqrt{x y}} + \frac{\sqrt{y}}{x + \sqrt{x y}})$

$x > 0; y > 0$

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Source: Vene T. Bogoslavov: Collection of Solved Problems #2 - 221.a.)

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$\frac{\sqrt{x} + \sqrt{y} - 1}{x + \sqrt{x y}} + \frac{\sqrt{x} - \sqrt{y}}{2 \sqrt{x y}} (\frac{\sqrt{y}}{x - \sqrt{x y}} + \frac{\sqrt{y}}{x + \sqrt{x y}})$

$= \frac{\sqrt{x} + \sqrt{y} - 1}{x + \sqrt{x y}} + \frac{\sqrt{x} - \sqrt{y}}{2 \sqrt{x y}} (\frac{\sqrt{y} (x + \sqrt{x y}) + \sqrt{y} (x - \sqrt{x y})}{(x - \sqrt{x y}) (x + \sqrt{x y})})$

$= \frac{\sqrt{x} + \sqrt{y} - 1}{x + \sqrt{x y}} + \frac{\sqrt{x} - \sqrt{y}}{2 \sqrt{x y}} \cdot [\frac{\sqrt{y} (x + \sqrt{x y} + x - \sqrt{x y})}{(x - \sqrt{x y}) (x + \sqrt{x y})}]$

= \frac{\sqrt{x} + \sqrt{y} - 1}{x + \sqrt{x y}} + \frac{\sqrt{x} - \sqrt{y}}{2 \sqrt{x y}} (\frac{2 x \sqrt{y}}{x^{2} - x y})

$= \frac{\sqrt{x} + \sqrt{y} - 1}{x + \sqrt{x y}} + \frac{\sqrt{x} - \sqrt{y}}{2 \sqrt{x y}} (\frac{2 x \sqrt{y}}{x (x - y)})$

= \frac{\sqrt{x} + \sqrt{y} - 1}{x + \sqrt{x y}} + \frac{\sqrt{x} - \sqrt{y}}{2 \sqrt{x y}} (\frac{2 \sqrt{y}}{x - y})

$= \frac{\sqrt{x} + \sqrt{y} - 1}{x + \sqrt{x y}} + \frac{\sqrt{x} - \sqrt{y}}{2 \sqrt{x} \sqrt{y}} (\frac{2 \sqrt{y}}{x - y})$

= \frac{\sqrt{x} + \sqrt{y} - 1}{x + \sqrt{x y}} + \frac{\sqrt{x} - \sqrt{y}}{\sqrt{x}} (\frac{1}{x - y})

= \frac{\sqrt{x} + \sqrt{y} - 1}{x + \sqrt{x y}} + \frac{\sqrt{x} - \sqrt{y}}{\sqrt{x}} (\frac{1}{x - y})

$= \frac{\sqrt{x} + \sqrt{y} - 1}{x + \sqrt{x y}} + \frac{\sqrt{x} - \sqrt{y}}{\sqrt{x}} \cdot \frac{1}{(\sqrt{x} - \sqrt{y}) (\sqrt{x} + \sqrt{y})}$

= \frac{\sqrt{x} + \sqrt{y} - 1}{x + \sqrt{x y}} + \frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x} + \sqrt{y}}

= \frac{\sqrt{x} + \sqrt{y} - 1}{x + \sqrt{x y}} + \frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x} + \sqrt{y}}

= \frac{\sqrt{x} + \sqrt{y} - 1}{x + \sqrt{x y}} + \frac{1}{x + \sqrt{x y}}

$= \frac{\sqrt{x} + \sqrt{y} - 1 + 1}{x + \sqrt{x y}}$

= \frac{\sqrt{x} + \sqrt{y}}{x + \sqrt{x y}}

$= \frac{\sqrt{x} + \sqrt{y}}{x + \sqrt{x y}} \cdot \frac{x - \sqrt{x y}}{x - \sqrt{x y}}$

= \frac{\sqrt{x} + \sqrt{y}}{x + \sqrt{x y}} \cdot \frac{x - \sqrt{x y}}{x - \sqrt{x y}}

$= \frac{x \sqrt{x} + x \sqrt{y} - x \sqrt{y} - y \sqrt{x}}{x^{2} - x y}$

= \frac{x \sqrt{x} - y \sqrt{x}}{x^{2} - x y}

$= \frac{\sqrt{x} (x - y)}{x (x - y)}$

$= \frac{\sqrt{x}}{x}$

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